Theory of Computing Systems

, Volume 56, Issue 4, pp 630–642

# New Lower Bounds on Circuit Size of Multi-output Functions

• Evgeny Demenkov
• Alexander S. Kulikov
• Olga Melanich
• Ivan Mihajlin
Article

## Abstract

Let Bn, m be the set of all Boolean functions from {0, 1}n to {0, 1}m, Bn = Bn, 1 and U2 = B2∖{⊕, ≡}. In this paper, we prove the following two new lower bounds on the circuit size over U2.
1. 1.
A lower bound $$C_{U_{2}}(f) \ge 5n-o(n)$$ for a linear function fBn − 1,logn. The lower bound follows from the following more general result: for any matrix A ∈ {0, 1}m × n with n pairwise different non-zero columns and b ∈ {0, 1}m,
$$C_{U_{2}}(Ax \oplus b)\ge 5(n-m).$$

2. 2.
A lower bound $$C_{U_{2}}(f) \ge 7n-o(n)$$ for fBn, n. Again, this is a consequence of the following result: for any fBn satisfying a certain simple property,
$$C_{U_{2}}(g(f)) \ge \min \{C_{U_{2}}(f|_{x_{i} = a, x_{j} = b}) \colon x_{i} \neq x_{j}, a,b, \in \{0,1\}\} +2n-\varTheta (1)$$
where g(f) ∈ Bn, n is defined as follows: g(f) = (fx1, … , fxn) (to get a 7no(n) lower bound it remains to plug in a known function fBn, 1 with $$C_{U_{2}}(f) \ge 5n-o(n)$$).

### Keywords

Circuit complexity Lower bounds Boolean functions Gate elimination

## Notes

### Acknowledgements

We would like to thank the anonymous reviewers for their comments that helped us to improve the readability of the paper.

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## Authors and Affiliations

• Evgeny Demenkov
• 1
• Alexander S. Kulikov
• 2
• Olga Melanich
• 2
• Ivan Mihajlin
• 2
1. 1.St. Petersburg State UniversitySt. PetersburgRussia
2. 2.St. Petersburg Department of Steklov Institute of MathematicsSt. PetersburgRussia